General Capelli-type identities

TL;DR

This paper introduces a new method to derive a broad family of Capelli-type identities, generalizing classical and higher identities, and confirming several conjectures in the theory of noncommutative determinants.

Contribution

A novel approach to derive a comprehensive family of Capelli identities, unifying and extending previous results and conjectures in noncommutative algebra.

Findings

Derived generalized Capelli identities including classical and higher cases

Established generalized Turnbull's identities for symmetric and antisymmetric matrices

Confirmed conjectures of Caracciolo, Sokal, and Sportiello on Capelli identities

Abstract

The classical Capelli identity is an important determinantal identity of a matrix with noncommutative entries that determines the center of the enveloping algebra of the general linear Lie algebra, and was used by Weyl as a main tool to study irreducible representations in his famous book on classical groups. In 1996 Okounkov found higher Capelli identities involving immanants of the generating matrix of $U(gl(n))$ which correspond to arbitrary orthogonal idempotent of the symmetric group. It turns out that Williamson also discovered a general Capelli identity of immanants for $U(gl(n))$ in 1981. In this paper, we use a new method to derive a family of even more general Capelli identities that include the aforementioned Capelli identities as special cases as well as many other Capelli-type identities as corollaries. In particular, we obtain generalized Turnbull's identities for both symmetric and antisymmetric matrices, as well as the generalized Howe-Umeda-Kostant-Sahi identities for antisymmetric matrices which confirm the conjecture of Caracciolo, Sokal, and Sportiello.